Distributed Mirror Descent Algorithm With Bregman Damping for Nonsmooth Constrained Optimization
Guanpu Chen, Gehui Xu, Weijian Li, Yiguang Hong
Abstract
To efficiently solve the nonsmooth distributed optimization with both local constraints and coupled constraints, we propose a distributed continuous-time algorithm based on the mirror descent (MD) method. In this article, we introduce the Bregman damping into distributed MD-based dynamics, which not only successfully applies the MD idea to the distributed primal-dual framework, but also ensures the boundedness of all variables and the convergence of the entire dynamics. Our approach generalizes the classic distributed projection-based dynamics, and establishes a connection between MD methods and distributed Euclidean-projected approaches. Also, we prove the convergence of the proposed distributed dynamics with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \mathcal {O}(1/t)$</tex-math></inline-formula> rate. For practical implementation, we further give a discrete-time algorithm based on the proposed dynamics with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \mathcal {O}(1/\sqrt{k})$</tex-math></inline-formula> convergence rate.